Proofs without words are drawings that 'prove' mathematical statements.For example, you could 'prove' that a(a + b) =
a^2 + ab by drawing the picture:
Can you see how that makes sense? (What property does it illustrate?)
What picture would prove what (a + b)^2 is equal to?
How about(a + b)(c + d)? (Extra: Can you think of another one to illustrate?)
When drawing pictures for your e-mail answer, choose a monospaced font and use spaces instead of tabs - since tabs have different sizes on different computers, using them can keep your drawing from lining up right.
Annie says: Solutions
Proofs without words are pretty neat, as they can provide a very clear explanation of algebraic concepts. (There are also good proofs that cover a lot of other areas of mathematics.)
Chui Yuk Man, from the Queen Elizabeth School in Hong Kong, provided a very thorough explanation of each part. Brian Gordon (one of those "adult" people) did a nice bonus part. I really like the solution from Tomomi Nakajima of Newport High School because it includes both the algebraic expansion and a well-labelled picture, which makes things very clear.
A lot of folks never did answer the first part - what property is illustrated by the example I gave? but I let them off the hook. It won't happen again! Brent Tworetzky tried to do (x-1)^2 as a bonus. His answer didn't quite illustrate it, but how would you do that? Also, check out Jessie Carr's solution - she did a really nice job without any pictures.
I especially like the answers that provide the algebraic expansion of the problem and then put the areas in the rectangles in the pictures. This makes it easy to compare the picture with the algebra and see that they match.
Brent Tworetzky tried to do (x-1)^2 as a bonus. His answer didn't illustrate it, but how would you do that?
A list of all the people who got this problem right and most of the solutions are also available.
Chui Yuk Man Grade: 7 School: Queen Elizabeth School, Hong Kong a(a+b) is the area of the whole picture. Consider it as one rectangle. a^2+ab is also the area of the whole picture but consider it as the sum of the area of a square and a rectangle. The following picture illustrates (a+b)^2=a^2+2ab+b^2 Area of whole picture = (a+b)^2 Area of big square = a^2 Area of small square = b^2 Sum of area of two rectangles = 2ab Area of whole picture = (a+b)^2 = a^2+2ab+b^2 a _______________________ | | | | | | | | | | | | a | | | | | | | | | |_______________|_______| | | | | | | b |_______________|_______| b Area of whole picture = (a+b)(c+d) Sum of area of four rectangles = ac+ad+bc+bd Area of whole picture=(a+b)(c+d) = ac+ad+bc+bd a b _________________________ | | | | | | c | | | | | | |_______________|_________| | | | d | | | |_______________|_________|
Brian Gordon Darthmouth '92 The given drawing represents the distributive property of multiplication over addition. Here's a drawing demonstrating that (a+b)^2=a^2 + 2ab + b^2: a b -------- | | | a | | | -------- | | | b -------- The squares have area a^2 and b^2, while the rectangles are ab each. The whole area is length times width, or (a+b)^2. For (a+b)(c+d), we again add areas of the separate rectangles to get ac+bd+ad+bc: a b ---------- | | | c ---------- | | | | | | | | | d | | | | | | ---------- A final drawing we could use would show that (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc: a b c --------------- | | | | a |-------------- |--|-|--------| b | | | | | | | | | | | | c the squares are along the diagonals, | | | | with pairs of each size rectangle. --------------- --bri
Tomomi Nakajima Grade: 9 School: Newport High School, Bellevue, Washington ______________________________ | | | From looking at the picture, | | | I noticed that the distributive | | | property is used: | | | a(a+b)=a^2+ab a | a^2 | ab | | | | | | | I can see how the picture | | | proves the equation is true. |_________________|____________| a b a b _____________________________ | | | The picture on the left proves that: | | | (a+b)^2=a^2+2ab+b^2 | | | a | a^2 | ab | This illustrates the distributive | | | property. | | | |_____________|_______________| | | | | | | b | ab | b^2 | | | | | | | |_____________|_______________| a b ______________________________ | | | The picture on the left proves that: | | | if a=b=c=d, then | | | (a+b)(c+d)=ac+ad+bc+bd c | ac | bc | | | | This illustrates the distributive | | | property. |_______________|______________| | | | | | | | ad | bd | d | | | | | | |_______________|______________| EXTRA: a b c ______________________________ | | | | The picture on the left proves that: | | | | (a+b+c)(d)=ad+bd+cd | | | | d | | | | This illustrates the distributive | | | | property. | | | | | | | | |________|_______|____________ |
Brent Tworetzky Grade: 10 School: P P Taravella High School, Broward County, Florida ------------- -------------- | | | | | | a | | | a| | | | | | | | | |-------------| |--------------| | | | | | | b | | | b| | | ------------- -------------- a b c d 2 (x - 1) -------------- | | | | | | | | | x | | | | | | |--------------| 1| | | -------------- 1 x
Jessie Carr Grade: 10 School: Germantown Academy, Fort Washington, PA Mrs.Carver's Geo.H The object of this problem was to prove something without using words.In the first problem (a+b)^2 I drew a square that was a by a and had an area of a^2. Then I added a piece to one side that was b by a and the area was ab. Then to the other side of the original square I added another piece like the one I just added. But in order to complete the diagram I had to add another piece that was b by b and had an area of b^2.When multiplied out that gave me a^2+2ab+b^2. In the second problem I did much the same thing as in the first. I made a rectangle that was a by d having an area of ad; then I added to that a rectangle that was b by d with an area of bd, then another rectangle that was c by a having an area of ac. Then the area that was left over had to be filled with a piece that was b by c with an area of bc. When the areas were all added they equaled ac+bc+ad+bd. This property was the distributive property. Another one that is possible would be if you had a(b+c): you would take a rectangle that was a by c and add to that one that was b by a, and when all multiplied out the area would be a(b+c).
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